A spacetime \((M, g)\)  is taken to be a paracompact, \(C^{\infty }\) , connected, Hausdorff manifold \(M\)  with a Lorentzian metric \(g\)  [1]}. So what is a classical singularity? A spacetime is by definition smooth, so `singular' points are not part of the spacetime; they must be cut out of the spacetime manifold. This leaves a `hole', with incomplete curves, a seeming boundary to spacetime. How do we complete spacetime, and how do we define a boundary \(\partial M\)  to spacetime? There have been a number of attempts, none of them entirely satisfactory. Note that Cauchy completeness works only in Riemannian metrics, not Lorentzian. Boundary definitions have included the a(abstract)-boundary of Scott and Szekeres [2]}, the b(bundle)-boundary of Schmidt [3]}, the c(causal)- boundary of Geroch, Kronheimer, and Penrose [4]} and the g(geodesic)-boundary of Geroch [5]}. In this discussion we will use Geroch's 1968 description of a classical singularity. He states that “a singularity is indicated by incomplete geodesics or incomplete curves of bounded acceleration in a maximal spacetime.” This is closest to the definition of classical singularity used in the famous singularity theorems of Hawking and Penrose, which predict that singularities are ubiquitous in exact solutions of Einstein's equations (see, e.g., [6]})).
